I've seen counters on some accounts. They've only made a few transactions, but counters are very big. What is the initial value of counter? How much will counter grow after I send a transaction?
The counter always increases by 1 for every operation - the difference is where the counter "starts".
When an implicit (tz*) account becomes active (goes from 0 balance to more than 0), it adopts the "global" counter. This is to save storage space, as when an implicit account is no longer active we don't need to track the counter anymore. It's important to note that an implicit account can go from inactive to active and back again many times.
Originated accounts start from 1 when they are originated.